Sunday, January 6, 2013

Before we start I'd like to thank listener SkepticHunter for posting another nice review on iTunes. Don't forget to post one too if you also like the podcast!

Recently I installed Google Earth on my iphone, and showed my daughter the cool feature where you can start with a view of the whole planet, and then zoom in to the exact spot where you currently are. Playing with this feature brought to mind an amusing essay I had recently read in Umberto Eco's "How To Travel With A Salmon", called "On The Impossibility of Drawing a Map of the Empire on a Scale of 1 to 1". The essay discusses a project of creating a map that is so detailed, it actually has a 1:1 scale, with each element on the map being the same size as the feature described.

The silly idea of a 1:1 scale map was apparently first proposed by Lewis Caroll in his lesser-known book Sylvie and Bruno, a somewhat enjoyable tale that didn't quite have the narrative flow of Alice in Wonderland, but shared its sense of absurdity. At one point, a character brags about his kingdom's mapmaking skills:What do you consider the largest map that would be really useful?"
"About six inches to the mile."
"Only six inches!" exclaimed Mein Herr. "We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!"
"Have you used it much?" I enquired.
"It has never been spread out, yet," said Mein Herr: "the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well."I think that quote pretty well speaks for itself.

Argentine writer Jorge Luis Borges was apparently a fan of Carroll's, and a few decades later in a short-short story commented on the idea himself. In Borges's vision, the 1:1 scale map is actually constructed, but then abandoned as useless, and left in western deserts to be gradually eroded away by the weather. In another of his essays, Borges points out that if England contains a perfect map of England, then that map must contain a depiction of the map itself, since that map is part of England-- and that depiction must include the depiction of the map, etc, out to infinity.

Eco's essay goes into much more detail than Carroll or Borges, discussing the major requirements for a 1:1 map to be useful. He insists that the map be able to exist within the country being mapped; I guess this is because it's not useful as a travel reference if the map must be elsewhere. It must be an actual map, rather than a mechanically created replica: for example, it would be cheating to take a plaster cast of the whole country and call it a map. The map also needs to be useful as a tool for referencing other parts of the country: so it can't be the case that for any location X, the depiction of X on the map lies on the portion of the map directly located at X, since then the map would be no more useful than a transparent sheet over the land. He also discusses various physical constraints such as the materials needed, folding techniques, and the effect on the actual country of having a giant map rolled out over it.

I also found an online discussion that pointed to several lesser-known references to such 1:1 scale maps. The rock band They Might Be Giants sings about a ship that is a 1:1 scale map of the state of Arkansas. The British comedy series "Blackadder" has an episode where an incompetent general is constructing a tabletop replica of recently conquered territory on a 1:1 scale. And modern comedian Steven Wright has a joke that goes "I have a map of the United States...actual size. It says, 'Scale: 1 mile = 1 mile.' I spent last summer folding it."

This whole discussion also seems to relate to the famous philosophical statement by Alfred Korzybski, "The map is not the territory". He was pointing out that if you are referring to a map, alternate view, or even a mental abstraction of something, you shouldn't confuse that with the thing itself. Naturally, this doesn't only refer to literal maps, but to just about anything you might see, interact with, or think about. This leads to another infinite regression problem, as we cannot actually sense physical things directly: we are always reacting to images on our retina, models in our mind, sets of beliefs we have about reality, or similar abstractions. So even when we think we are directly interacting with actual reality, we are actually dealing with somewhat less fidelity than the ideal 1:1 map.

But, in any case, what I consider the most ironic thing about all these riffs on Lewis Carroll's original attempt at absurdity is that the whole concept of 1:1 scale maps is no longer so absurd. Modern software tools like Google Earth really do allow us to depict arbitrary scales, even 1:1, by storing the map virtually and letting us just zoom in on the parts we currently need. Technically I don't think we have quite the satellite resolution to consider Google Earth maps to be 1:1 scale, but we're getting pretty close. And I'm sure philosophers will continue to point out that the Google Map is not the territory-- but for practical purposes, it seems close enough to me. I wonder what other ridiculous absurdities from Alice in Wonderland or Sylvie and Bruno will be rendered mundane by future technologies.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

The Math Mutation Book

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